The harmony of being at the minimum

I am always amazed by the idea of being at the minimum. Because every physical system near the minimum is an harmonic oscillator, and we love harmonic oscillator !

Let us consider a generic potential energy function U(x) and suppose that x_0 is a stationary point for U(x). A stationary point could be a minimum, maximum or inflection point of a differentiable function, in which the first derivative is null. In other words in a stationary point the function stops to increase or decrease.

If we expand in Taylor series the potential U(x) in x_0 to the second order we have:

U(x) = U(x_0) + \frac{1}{2}\frac{d}{dx}U(x=x_0) (x-x_0) + \frac{d^2}{dx^2}U(x=x_0) (x-x_0)^2 + o(x-x_0)^3

Since x_0 is a stationary point for U(x) the first derivative is null \frac{d}{dx}U(x=x_0) = 0 and the potential becomes:

U(x) = U(x_0) + \frac{1}{2}\frac{d^2}{dx^2}U(x=x_0) (x-x_0)^2 + o(x-x_0)^3

The first term U(x_0) is a number equal to the value of the potential in the stationaty point. Since a generic potential energy is defined up to an additive constant, we can put U(x_0)=0. Furthermore, the expression o(x-x_0)^3 means if we stop the expansion of the potential to the second order we commit an error that is an infinitesima quantity lower than (x-x_0)^3 .

Finally, we can express the potential energy near a stationary point in this form:

U(x) = \frac{1}{2}k (x-x_0)^2

(where k=\frac{d^2}{dx^2}U(x=x_0)) that is the potential energy of a harmonic oscillator !

The harmonic oscillator is wherever you look in theoretical Physics (Quantum Mechanics, Quantum Field Theory , Solid State Physics and so on …).
The most important thing is we are able to exactly solve the harmonic oscillator, and then you can solve every physical system near a stationary point.

Lennard - Jones Energy Potential

Lennard – Jones Energy Potential (source: Wikipedia)

For example, consider the Lennard – Jones potential, phenomenologically describing the short range interaction between neutral particles:


where \epsilon is the depth of the potential,  r is the particles relative distance and r_m is the stationary point. If we expand the LJ potential near the minimum we have:


(where  k=\frac{-72\epsilon}{r_0^4} )

When the inter-particle distance is approximately equal to r_m they vibrate in harmony !

So, if you stay in the minimum of your potential energy, sin prisa, you can vibrate sin pausa (cit. R.B)


One comment

  1. Cool 🙂

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